Do I have to turn over money in a bank account shared between my now deceased father and I? Or is the money mine now?

The presumption is that a joint account belongs to the surviving joint account holder. That presumption can be overcome if it can be shown that the other holder was just put on for convenience. That is usually shown by showing that all the funds in the account came from the older person. In your case, you can apparently show that you put your own funds in the account also, which gives you a stronger case. In any event, the trustee has no authority over the account, since it was not in the trust. If the trustee is also the executor, she may attempt to fight to gain control of the account. You have to decide whether you want to fight about this. If it had all been your dad's money, I would have encouraged you to voluntarily split the funds with other beneficiaries.